Beam Modal Analysis

1

VIA University College

Mechanical Engineering

Programme

Dorin Bordeasu (164631)

Khem Raj Guatam (164645)

Javier Camacho (164649)

Beam Modal Analysis

Spring 2013 Beam Modal Analysis Page 2

Table of Content

Introduction ................................................................................................................. 4

General Project Purpose and Planning .......................................................................... 5

Mathematical Models .................................................................................................. 6

Displacement Static Analysis ........................................................................................ 6

Diagrams ....................................................................................................................... 7

Transversal Displacement Stiffness Function Analysis ................................................ 9

2D Transversal Displacement Stiffness ........................................................................ 9

3D Transversal Displacement Stiffness ........................................................................ 9

Angular Displacement Stiffness Function Analysis .....................................................10

2D Angular Displacement Stiffness ............................................................................10

3D Angular Displacement Stiffness ............................................................................10

Total Displacement Stiffness Function Analysis .........................................................11

2D Total Displacement Stiffness ................................................................................11

3D Total Displacement Stiffness ................................................................................11

Euler - Bernoulli Method..............................................................................................12

Finite Element Method ................................................................................................19

2 Elements Analysis ....................................................................................................20

4 Elements Analysis ....................................................................................................21

CAD Finite Element Model .......................................................................................... 22

2D Simplified Model ....................................................................................................22

Model Information .......................................................................................................22

Parametric Settings .....................................................................................................22

Boundary Conditions ...................................................................................................22

Mesh statistics ............................................................................................................23

Modal Analysis ............................................................................................................24

3D Model .....................................................................................................................25

Boundary Conditions ...................................................................................................25

Modal Analysis ............................................................................................................26

FEM Results Analysis ................................................................................................. 27

Convergence Analysis .................................................................................................27

Frequency Analysis ......................................................................................................29


Incremental Number of Elements Analysis .................................................................31

Results Comparison of FEM to Beam Theory..............................................................32

Experimental Modal Analysis ...................................................................................... 33

Problem Definition ..................................................................................................... 34

Experiment Design ..................................................................................................... 34

Data Gathering .......................................................................................................... 39

Experimental Uncertainty Analysis .............................................................................. 40

Measurement Elemental Errors Sources ....................................................................40

Systematic Errors ........................................................................................................41

Random errors ............................................................................................................41

Random Errors Uncertainty .........................................................................................42

Systematic Error Uncertainty.......................................................................................42

Total Uncertainty (Simplification) ................................................................................42

Probability Analysis .................................................................................................... 43

Probability Distribution Function Analysis ...................................................................44

Experiment Result Analysis......................................................................................... 45

Results Comparison ....................................................................................................45

Conclusions ............................................................................................................... 47

List of Reference ........................................................................................................ 48

Appendix ................................................................................................................... 49

Appendix 1: Static Analysis .........................................................................................49

Appendix 2: Euler – Bernoulli Beam Modal Analysis ..................................................49

Appendix 3: Finite Element Method ............................................................................49

Appendix 4: Probability Analysis..................................................................................49

Appendix 5: Uncertainty Analysis ................................................................................49


Introduction

Analysing the natural frequency of system is fundamental for structural and acoustic

design in order to predict and understand system’s dynamic behaviour.

All objects can be regarded as spring. When small external forces try to disturb them they

will try to get back to its original position because of stiffness of spring. When external

forces act, spring will tend to come to same position and when this process repeats

vibration is created. Vibration thus created is very small and cannot be detected easily.

As the disturbing frequency becomes larger the mass effect of system will oppose motion.

In order for system to vibrate external forces should be very big to overcome inertia of

mass to make rapid changes in direction of motion. In this case effect of stiffness of spring

is so small it becomes negligible.

As explained above, we can easily see system has two different region of behaviour. First

when small external forces are acted spring stiffness tries to keep the mass at equilibrium

position and system is stiffness controlled. In other words spring stiffness will basically act

as inward force as it will try to restore mass to base position.

And in second region of behaviour, when system is mass controlled, the inertia of mass will

try to keep the mass in same direction opposing rapid change of motion in extreme end of

each stroke. Thus mass effect can be explained as outward one.

Stiffness effect is independent of disturbing frequency but mass effect for given amplitude

is square function of frequency. So when frequency starts increasing from zero at certain

frequency stiffness effect and mass effect cancels out. At that point neither factor

restrains movement of mass. So system goes wild and vibrates maximum without control.

The frequency at which this behaviour occurs is called natural Frequency of a system.

As a course work for FEM and EEX natural frequency of beam structure were calculated

with different approaches and result were analysed. For simplicity, natural frequency of a

clamped-clamped circular bar made out of structural steel was the subject for experiment.


General Project Purpose and Planning

The purpose of this project is to perform a Beam Modal Analysis in order to find at least

the lowest natural frequency of a Clamped- Clamped beam system by different methods

and software and compare the results with experiment data gathered.

The diagram below describes the systematic procedure.


Mathematical Models

Displacement Static Analysis


Diagrams



Transversal Displacement Stiffness Function Analysis

2D Transversal Displacement Stiffness

3D Transversal Displacement Stiffness


Angular Displacement Stiffness Function Analysis

2D Angular Displacement Stiffness

3D Angular Displacement Stiffness


Total Displacement Stiffness Function Analysis

2D Total Displacement Stiffness

3D Total Displacement Stiffness


Euler - Bernoulli Method








Finite Element Method

Finite element theory is used to find natural frequency of given beam. Using Euler’s

governing equation and finite element concepts as described in appendix (Global

matrices) we can formulate global mass and stiffness matrixes of one element as

Using these values of mass and stiffness and comparing to the equation, natural

frequency can be calculated.


2 Elements Analysis

As the subject is clamped - clamped the analysis could not be performed with one

element; because we will have zero degree of freedom system.

Using two beam elements and creating energy equation for both elements and adding

those equations we get

Solving the Eigen-value problem shown above, two natural frequencies were obtained as

91.32 and 329.266 Hz respectively. First natural frequency thus obtained is close to

analytic result by the Euler’s method but second frequency was not predicted correctly. As

general thumb rule to predict nth natural frequency by FEM at least 2n degree of freedom

system should be created. Please refer appendix (FEM) for definition of symbols used


4 Elements Analysis

In this case the Beam was treated as 4 element system and similar procedure was

performed. That way beam had 6 degree of freedom and 6x6 matrices were formed for

global mass and stiffness matrixes. Final equation thus becomes:

Where l is of course length of each element in our case (1/4)

Solving this Eigen value problem six frequencies obtained are

As expected first 3 natural frequency thus obtained were close to analytical results. More

elements and more degree of freedom system are generally used by FEM software like e

ANSYS and COMSOL thus predicting more precise result.


CAD Finite Element Model

2D Simplified Model

Model Information

The model is defined as a line with a circular cross section; it is used in order to simplify

the computational requirement.

Parametric Settings

Name Value Unit

Density 7850 kg/m^3

Poisson's ratio 0.31 -

Young's modulus 200e9 Pa

Section type Circular

Diameter 0.02 m

Length 1 m

Boundary Conditions


Mesh statistics

In the 2D model the range of elements it is set up between 2 until 3000 elements.


Modal Analysis

Mode #1

89.628 Hz

Mode #2

246.41 Hz

Mode #3

481.34 Hz

Mode #4

792.11Hz


3D Model

To be close to real situation the bar mentioned in 2 D analysis was welded to two pieces of

rectangular steel (250x200mm) pieces. Analysis was carried assuming the bottom and top

part of rectangular piece to be fixed and using 50 elements

Boundary Conditions


Modal Analysis

Mode #1

88.235 Hz

Mode #2

242.54 Hz

Mode #3

474.03 Hz

Mode #4

780.34 Hz


FEM Results Analysis

Convergence Analysis

The analysis was performed to find the minimum number of elements required in order to

get the most accurate results regarding efficiency of the number of computations.

The convergence analysis yield as a result in this case that COMSOL (start converge at 10

elements) and ANSYS start converge at 15 elements).

As a conclusion, in COMSOL the values start converge early in number of elements than

ANSYS. That means in this case is required less number of elements which affect directly

in the matrix dimension required to compute, reducing the number of it computations and

the resources usage from the PC, in this case COMSOL is the most efficient software.



Frequency Analysis


This diagrams yields clearly the different between analytical method, finite elements

method (manually) and CAD Finite element.

The required accuracy of the results will depend on the minimum requirements of the

structures application.


Incremental Number of Elements Analysis

In this case the group decided to perform an analysis of the higher number of elements

increment (100000 elements), as the figures shows the behaviour of the system is

completely different than the real situation. The beam is getting stiffer as much the

number of elements is increased from the last converged amount of elements. That is why

it is important to take into account the convergence analysis in order to perform efficient

and accurate computations.

Mode#1

131.7613 Hz

Mode#2

573.6874 Hz

Mode#3

852.1884 Hz

Mode#4

971.9279 Hz


Results Comparison of FEM to Beam Theory

In general, the displacement evaluated by FEM using the cubic function approximation for

δ(x), is lower than those of the beam theory except at the nodes. This is true for beams

subjected to some form of distributed load that are modelled using the cubic displacement

approximation function. The exception to this result is at the nodes, where the beam

theory and FEM results are identical because of the work – equivalent discrete loads at

the nodes.

The beam theory solution predicts a quartic (fourth – order) polynomial expression for

beam subjected to uniformly distributed loading, while the FEM solution δ(x) assumes a

cubic (third – order) displacement behaviour in each beam all load conditions(Under

uniformly distributed loading conditions, the beam theory solutions predicts a quadratic

moment and linear shear force in the beam. The FEM solution using the cubic

displacement function predicts a linear bending moment and a constant shear force within

each element used in the model.).

The FEM solution predicts a stiffer structure than the actual one. However, as more

elements are used in the model, the FEM solution converges to the beam theory solution.

The group has to mention that until this part of the entire project (which is basically a

simple structure) that the most relevant knowledge acquired is how powerful FEM can be

especially in the computation of complex structures or designs, but requires a real

understanding of the limitations of the method itself, or may cause serious errors.


Experimental Modal Analysis


Problem Definition

As it is already stated in the beginning of this project the main purpose of this experiment

it is to find at least the lower natural frequency of the fixed-fixed beam, which is the most

important considering structural design and the phenomenal behaviour of it when reach

that frequency(resonance).

The group has to emphasize in this point because regarding the complexity of a real

situation could be expected that to find a higher natural frequencies it is not be possible to

perform.

Experiment Design

Experimental Approach

To define the experimental approach brain storming was performed in order to find the

most suitable experimental design regarding simplicity, sources availability and cost.


Analytical Model

The analytical model is performed on the first part of this report and compared with the

CAD model. This model will be use full to analyse the data gather on the experiment.

Measured Variables

Considering the nature of the experiment and the purpose of it, the most relevant variable

which the experiment will evaluate is the natural frequency at different values of inputs.

It is expected that the natural frequency does not change over the different inputs; of

course the amplitude will change but in this case it not relevant.

Instrument Selection

No. Apparatus Manufacture Model Serial Nr.

1 Accelerometer Brüel & Kjær · Denmark · 4391 1721031

2 Kistler Charge Meter Kistler Instrumens 5015A1001 1433047

3 Data Translation Adept Turnkey Pty DT9804ECIBNC-80I 615511

4 Variable frequency Filter Ap. Circuit AP-255-5 86260

5 Data Translation scope 2.2.0.30 Data Translation, Inc.



Experiment Setup Procedure

1. The structure was placed in the heavy table and fixed with help of clamping

tools in both the end. The procedure was performed to minimize any damping.

2. A Transducer sensor (type 4391 industrial), was mounted to the steel rod at

distance 0.5m from each ends. The magnetic cap in the sensor was used to

position it correctly at surface of the rod.

3. The sensor was connected to an universal charge amplifier ( Kistler Charge

Meter type 5015), which amplifies the signal from the piezoelectric sensor,

display values on the screen or sent it further. The amplifier was set to

fallowing settings

Unit: acceleration in mm/s^2

Signal evaluation: mean

Sensitivity 1pc/m/s^2

Filter 0.1s

Lp filter: 2 kHz

Output: +- 10 V

4. Although charge meter has in build filter and can read some data on its own, it

was used just as signal reader and was connected to the variable frequency

filter to filter which was set to be used as low pass filter With fallowing settings

Mode: low pass filter

Damping/normal: Normal

Output: 50*10Hz

5. The Data Translation DT9804 is connected to AC channel 0 to receive the

filtered signal, which analyze the system data, processes it and send it further

trough the USB cable to a computer.

6. The Pc is connected to Data Translation DT9804, and the software, Data

Translation scope Version 2.2.0.30 is used to plot FFT signals with the voltage

on the y axis and the frequency on the x axis.

7. An impact hammer with soft impact area is used to excite the beam

8. Experiment was performed several times

9. The Data were processed in PC



Data Gathering


Experimental Uncertainty Analysis

Measurement Elemental Errors Sources

Our system is identified as second order dynamic system. Various devices are used to

perform the experiment so we have many sources of errors. Some expected sources of

errors in each device is explained in the chart.


Systematic Errors

Zero error

Calibration error or zero error is the systematic error caused by wrong Calibration of

measuring instruments. This kind of errors can be corrected to some extend my

calibrating instrument as precise as possible but some residual errors will still

persist. They are also caused because relation between input and output in the

instrument is not always linear. In special case of ours we have this kind of errors in

sensors and charge amplifier. Amplifier manufacturer has provided some data

about uncertainty caused by zero error but it is difficult to find data for sensor as

the error depends not only in instrument but also the way it will be used.

Insertion errors

This kind of error is cause because measurement device may not be correctly

placed. In our case flat magnetic transducer sensor may not be mounted tangential

to surface of the circular beam, whose natural frequency is to be measured.

Errors from surrounding

Experiment we are performing is likely to be influenced by surrounding noises.

Random errors

The errors which occur while taking repeated measurement with same parameters

are random errors. As our experiment is surrounded by multiple electric and

magnetic fields this kind of errors are certain. Amplifier may produce different data

in different measurement because of change of temperature in components over

time. Consideration will be taken properly to minimize this kind of errors.

Accuracy errors:

This error comes from accuracy of measurement by instrument. Instrument we are

using have certain percentage of accuracy for specific range. Instruments will be

used in the range where these errors can influence our result minimum.

Hysteresis:

In experiment same value of measurand, can be read differently depending if

reading was taken when values were increasing or decreasing. This phenomenon

occurs because of friction, mechanical flexure or electric capacitance. If reading

could be taken always at same point this error can be treated as systemic error but

since that cannot be done it can be classified as random error.

Resolution error:

As multiple instruments will be used, it is expected instruments will make reading in

discrete steps instead of continuous, resulting these kind of errors. Incorrectly

readability of outputs is also kind of resolution error.


Linearity errors

Output is expected to be proportional to input .If systematic errors at different point

is known calculated value must satisfy general relation between input and output.

But that may not be accurate or constant value. This deviation is linearity error.

Other errors

Thermal errors and sensibility errors can be both cause systematic and random

errors.

Random Errors Uncertainty

Uncertainty due to random error is computed from the standard deviation of a sample

data.

Systematic Error Uncertainty

Uncertainty due systematic error, UBias , is based on past experience, manufacture

specifications or other information.

In most of the cases, this uncertainty estimate is assumed to be roughly equivalent to a

95% confidence interval or limits for a systematic error.

Total Uncertainty (Simplification)


Probability Analysis

Sample and SD Interval

Confidence Interval on Mean


Probability Distribution Function Analysis

Normal Probability Density Function

Probability Prediction of the Confidence Interval

P(z1≤z≤z2)=36.89%

Probability Prediction of the Mean and SD Interval

P(z1≤z≤z2)=68.27%


Experiment Result Analysis

Results Comparison



Conclusions

The charts shows that the experimental result differs less than 3.5 % from result obtained

from CAD software and 4 % with analytical results obtained from the beam theory. Thus

theory and results from experiment agree reasonably well.

The lower value of frequency obtained from experiment can be explained as the difference

in boundary condition of real situation. Analytical calculation and CAD simulation assumes

to have perfect clamped- clamped boundary, which defines system to be stiffer than the

real model. Real model was clamped, but possibility of slight movement existed.

The total uncertainty is located in the confidence interval (of 95%) between (86.436

86.606) with a 33.46 % of probability of occurrence in the experiment. The experiment

was repeated and replicated but of course there is place for many improvements.

It has to be mentioned that the project provided very good understanding of the theory,

use of computer software and the experimental procedures. It also gave insight to how

different approach can be used to find the same result.

Computer tools can be powerful in computation of complex structure but requires a real

understanding and limitation of method itself. Real life situation will always vary from

theory to certain extend and those consideration must be made. Use of Computer software

without understanding the real situation can be very dangerous and experimental test

should often verify the results unless the theory is understood clearly.

The subject of project was simple structure, none the less it required a lot of self-studying

in many fields and required deeper mathematical understanding. But under pressure is

when you stretch and put yourself in test. The group believes it was successful in that test.


List of Reference

1. Meriam, J. L., 2006, Engineering Mechanics: Static. 6th ed.

2. Boresi, A. P. and Schmidt R. J, 2009, Advanced mechanics of materials,, Wiley

3. Hibbeler, R. C., 2010, Engineering Mechanics Dynamics, 12th ed. Prentice Hall

4. Hibbeler, R. C., 2011, Mechanics of Materials, 8th ed. Prentice Hall

5. Engineering Vibrations Daniel J. Inman

6. Vibration Simulation using Matlab and ANSYS Michael R. Hatch

7. Advanced Modern Engineering Mathematics 4th Glyn James

8. Mathematical Methods for Mechanics Eckart W. Gekeler

9. Introduction to Engineering Experimentation Anthony J. Wheeler

10. Matlab

11. Mathcad

12. Inventor

13. ANSYS

14. COMSOL

15. E. and F.N spoon,1991,3rd edition, Noise control in industries

16. Inman, D J, 3rd edition, Engineering Vibration

17. Mathur, D.S 2005,Mechanics

18. Various Internet journal


Appendix

Appendix 1: Static Analysis

Appendix 2: Euler – Bernoulli Beam Modal Analysis

Appendix 3: Finite Element Method

FEM Calculations

Global Mass and Stiffness Matrix

Appendix 4: Probability Analysis

Appendix 5: Uncertainty Analysis

Documents

Beam Modal Analysis